Math 360, Fall 2014, Assignment 3

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.

- Mary Shelley, Frankenstein

Carefully define the following terms, then give one example and one non-example of each:

1. Isomorphism (from one binary structure to another).
2. Isomorphic (binary structures).
3. Structural property (of a binary structure).
4. Identity element.
5. Inverse (of an element of some binary structure with an identity).
6. Group.
7. Abelian group.

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning the uniqueness of identity elements (Theorem 3.13).
2. Theorem concerning the uniqueness of inverses in groups (second part of Theorem 4.17).

Solve the following problems:

1. Section 3, problems 2, 4, 8, 9, and 17.
2. Section 4, problems 1, 5, 7, 12, 13, and 17.

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Solutions:

In an effort to promote collaboration, I will only post work on problems whose problem numbers are elements of $[2]_{3}$. Please join me in working on the problems here. -Ian

Definitions:

3. A structural property of a binary structure $(S,\ast)$ is a property that holds for all structures that are isomorphic to $(S,\ast)$.
6. A group $(G,\ast)$ is a binary structure for which the following three properties hold:
   1. $\ast$ is associative
   2. $G$ contains an identity element $e$, i.e. for all $a\in G$, $a\ast e = e\ast a = a$
   3. For all $a\in G$, there exists $a'\in G$ with $a\ast a'=a'\ast a=e$

Theorems:

2. In any group $(G,\ast)$, for each $a\in G$, there is only one element $a'\in G$ with $a\ast a'=a'\ast a=e$, where $e$ is the identity element.

Problems:

Section 4

1)Not a group. $(\mathbb{Z},\ast)$ fails to have an inverse at $0$.

5)Not a group. Ordinary division is not associative.

7)$(\mathbb{Z}_{1000},+_{1000})$

12)Not a group. Any matrices with a zero along the diagonal fail to have an inverse.

13)Is a group. The multiplication of two $n \times n$ diagonal matrices with nonzero diagonal entries yields an $n \times n$ diagonal matrix with nonzero diagonal entries, matrix multiplication is associative, the Identity Matrix is an identity, and any such matrices will have non-zero determinants, meaning they will have inverses (diagonal with nonzero diagonal entries.(.

17)Is a group. Multiplying two $n \times n$ upper-triangular matrices with determinant 1 gives another $n \times n$ upper-triangular matrix with determinant 1. Matrix multiplication is associative. The Identity Matrix is upper-trianguler and has determinant one, and the inverse of an $n \times n$ upper-triangular matrix with determinant 1 will also be an $n \times n$ upper-triangular matrix with determinant 1.